Root of unity

An $N$th root of unity is a complex number $z$ satisfying $z^N=1$. The primitive root $e^{2\pi i/N}$ generates all $N$th roots.

The Ramanujan sum is $c_n(k)={\sum }_{1\le j\le n,(j,n)=1}{e}^{2\pi ijk/n}$. It is an integer: insert $1_{(j,n)=1}={\sum }_{d\mid (j,n)}\mu (d)$, sum the geometric series, and obtain $c_n(k)={\sum }_{r\mid (n,k)}r\mu (n/r)$. Here $\mu$ is the Möbius function. Such sums connect roots of unity with integer character of a representation values.

A primitive $n$th root of unity has multiplicative order exactly $n$.

For $\omega =e^{2\pi i/N}$,This finite geometric sum is the orthogonality relation underlying the discrete Fourier transform.

For $\omega =e^{2\pi i/N}$,This is the finite geometric series identity and expresses orthogonality of the characters of ${\mathbb{Z}}_N$.